Coloring rules for finite trees, and probabilities of monadic second order sentences

Abstract

A system of coloring rules is a set of rules for coloring the vertices of any finite rooted tree, starting at its leaves, which has the property that the color assigned to each vertex depends only on how many of its immediate predecessors there are of each color. The asymptotic behavior of the fraction of n vertex trees with a given root color is investigated. As a primary application, if φ is any monadic second-order sentence, then the fractions μn(φ), νn(φ) of labeled and unlabeled rooted trees satisfying φ, converge to limiting probabilities. An extension of the idea shows that for a particular model of Boolean formulas in k variables, k fixed, the fraction of such formulas of size n which compute a given Boolean function approaches a positive limit as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 453–485, 1997